 
(* ::Section:: *)
(* InverseMellin *)
(* ::Text:: *)
(*InverseMellin[exp, y] performs the inverse Mellin transform of polynomials in OPEm. The inverse transforms are not calculated but a table-lookup is done. WARNING: do not "trust" the results for the inverse Mellin transform involving SumT's; there is an unresolved inconsistency here (related to $(-1){}^{\wedge}m).$.*)


(* ::Subsection:: *)
(* See also *)
(* ::Text:: *)
(*DeltaFunction, Integrate2, OPEm, SumS, SumT.*)



(* ::Subsection:: *)
(* Examples *)



InverseMellin[1/OPEm,y]

InverseMellin[1/(OPEm+3),y]

InverseMellin[1,y]

InverseMellin[1/OPEm^4,y]

InverseMellin[1/OPEm+1,y]

InverseMellin[1/i+1,y,i]


(* ::Text:: *)
(*The inverse operation to InverseMellin is done by Integrate2.*)


Integrate2[InverseMellin[1/OPEm,y],{y,0,1}]


(* ::Text:: *)
(*Below is a list of all built-in basic inverse Mellin transforms .*)


list={1,1/(OPEm+n),1/(-OPEm+n),PolyGamma[0,OPEm],SumS[1,-1+OPEm],SumS[1,-1+OPEm]/(OPEm-1),SumS[1,-1+OPEm]/(1-OPEm),SumS[1,-1+OPEm]/(OPEm+1),SumS[1,-1+OPEm]/OPEm^2,SumS[1,-1+OPEm]/OPEm,SumS[1,-1+OPEm]^2/OPEm,SumS[2,-1+OPEm],SumS[2,-1+OPEm]/OPEm,SumS[3,-1+OPEm],SumS[1,1,-1+OPEm],SumS[1,OPEm-1]^2,SumS[1,2,-1+OPEm],SumS[2,1,-1+OPEm],SumS[1,-1+OPEm]^3,SumS[1,-1+OPEm] SumS[2,-1+OPEm],SumS[1,1,1,-1+OPEm]};
im[z_]:=z \[LongRightArrow]InverseMellin[z,y]
im[OPEm^(-3)]

im[OPEm^(-2)]

im[PolyGamma[0,OPEm]]

im[SumS[1,OPEm-1]]

im[SumS[1,OPEm-1]/(OPEm-1)]

im[SumS[1,OPEm-1]/(OPEm+1)]

im[SumS[1,-1+OPEm]/OPEm^2]

im[SumS[1,-1+OPEm]/OPEm]

im[SumS[1,-1+OPEm]^2/OPEm]

im[SumS[2,OPEm-1]]

im[SumS[2,OPEm-1]/OPEm]

im[SumS[3,OPEm-1]]

im[SumS[1,1,OPEm-1]]

 im[SumS[1,2,OPEm-1]]

 im[SumS[2,1,OPEm-1]]

im[SumS[1,1,1,OPEm-1]]

Clear[im,list];